Solution to classify product names
If you're looking for an existing solution, the best approach I found was using a TF-IDF model, check out the links below which have similar examples which should be easily adapted for your dataset.

https://www.kaggle.com/selener/multi-class-text-classification-tfidf#targetText=Text%20classification%20(multiclass)&amp;targetText=With%20the%20aim%20to%20classify,one%20of%20the%20product%20categories).

https://github.com/susanli2016/Machine-Learning-with-Python/blob/master/Consumer_complaints.ipynb

However if you specifically want to go for a DNN approach, there are a few options you can take for a multi-class text classification. Try looking into a simple CNN classifier, which is a relatively lightweight approach, computationally speaking, yet showing pretty good results:

https://medium.com/jatana/report-on-text-classification-using-cnn-rnn-han-f0e887214d5f

Alternatively, you can use a word2vec or a doc2vec model to map your sentences to unique vectors, and then put them through a regression algorithm.

https://towardsdatascience.com/multi-class-text-classification-model-comparison-and-selection-5eb066197568